Playing with Numbers

Q1: Multiple Choice Type

i. Which of the following number(s) are divisible by 2:

Rule: A number is divisible by 2 if its last digit is even.
192 → last digit = 2 ✔
1660 → last digit = 0 ✔
1101 → last digit = 1 ✘
Answer: a and b only

ii. Which of the following number(s) are divisible by 3:

Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.
261 → 2+6+1 = 9 ✔
777 → 7+7+7 = 21 ✔
6657 → 6+6+5+7 = 24 ✔
Answer: a, b, and c

iii. Which of the following number(s) are divisible by 6:

Rule: A number is divisible by 6 if it is divisible by both 2 and 3.
2292 → even ✔ and 2+2+9+2 = 15 → divisible by 3 ✔
2612 → even ✔ and 2+6+1+2 = 11 → not divisible by 3 ✘
5940 → even ✔ and 5+9+4+0 = 18 → divisible by 3 ✔
Answer: a and c only

iv. Which of the following number(s) are divisible by 9:

Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
1863 → 1+8+6+3 = 18 ✔
2064 → 2+0+6+4 = 12 ✘
6111 → 6+1+1+1 = 9 ✔
Answer: a and c only

v. Which of the following number(s) are divisible by 4:

Rule: A number is divisible by 4 if the last two digits form a number divisible by 4.
360 → last two digits = 60 → 60 ÷ 4 = 15 ✔
4093 → 93 ÷ 4 = 23.25 ✘
5348 → 48 ÷ 4 = 12 ✔
Answer: a and c only

vi. Which of the following number(s) are divisible by 5:

Rule: A number is divisible by 5 if it ends in 0 or 5.
3250 → ends in 0 ✔
5557 → ends in 7 ✘
39255 → ends in 5 ✔
Answer: a and c only

vii. Which of the following number(s) are divisible by 10:

Rule: A number is divisible by 10 if it ends in 0.
5100 → ends in 0 ✔
4612 → ends in 2 ✘
3400 → ends in 0 ✔
Answer: a and c only

viii. Which of the following number(s) are divisible by 11:

Rule: Alternating digit sum (odd-even-odd…) difference should be divisible by 11.
2563 → 2 – 5 + 6 – 3 = 0 ✔
8307 → 8 – 3 + 0 – 7 = -2 ✘
95635 → 9 – 5 + 6 – 3 + 5 = 12 ✘
Answer: a only

Q2: Show that the number 21952 is divisible by 7.

Divisibility Rule of 7: Double the last digit, subtract it from the rest of the number. Repeat the process until you get a small number. If the final number is divisible by 7, then the original number is also divisible by 7.
Step 1: Last digit of 21952 is 2. Double it: 2 × 2 = 4
Subtract 4 from 2195: 2195 − 4 = 2191
Step 2: Last digit of 2191 is 1. Double it: 1 × 2 = 2
Subtract 2 from 219: 219 − 2 = 217
Step 3: Last digit of 217 is 7. Double it: 7 × 2 = 14
Subtract 14 from 21: 21 − 14 = 7
Step 4: Final result is 7, which is divisible by 7.
Answer: Yes, 21952 is divisible by 7 (Final result = 7).

Q3: Show that the number 6486 is not divisible by 7.

Divisibility Rule of 7: Double the last digit and subtract it from the rest of the number. Repeat until you reach a small number. If the final number is divisible by 7, then the original number is also divisible by 7.
Step 1: Last digit of 6486 is 6. Double it: 6 × 2 = 12
Subtract 12 from 648: 648 − 12 = 636
Step 2: Last digit of 636 is 6. Double it: 6 × 2 = 12
Subtract 12 from 63: 63 − 12 = 51
Step 3: 51 is not divisible by 7.
Step 4: Therefore, 6486 is not divisible by 7.
Answer: No, 6486 is not divisible by 7 (Final result = 51).

Q4: For what value of digit x is:

i. 1×5 divisible by 3?

Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.
Sum = 1 + x + 5 = (x + 6)
We need x + 6 divisible by 3.
Try x = 0 to 9:
x = 0 → 6 ✔
x = 3 → 9 ✔
x = 6 → 12 ✔
x = 9 → 15 ✔
Answer: x = 0, 3, 6, 9

ii. 31×5 divisible by 3?

Sum = 3 + 1 + x + 5 = (x + 9)
x + 9 divisible by 3 → x = 0, 3, 6, 9
Answer: x = 0, 3, 6, 9

iii. 28×6 a multiple of 3?

Sum = 2 + 8 + x + 6 = (x + 16)
x + 16 divisible by 3 → x = 2, 5, 8
Answer: x = 2, 5, 8

iv. 24x divisible by 6?

Rule: Must be divisible by both 2 and 3.
Divisible by 2 → last digit x must be even: 0, 2, 4, 6, 8
Sum = 2 + 4 + x = x + 6 → divisible by 3 → x = 0, 3, 6, 9
Common values: x = 0, 6
Answer: x = 0, 6

v. 3×26 a multiple of 6?

Last digit = 6 ✔ (even)
Sum = 3 + x + 2 + 6 = x + 11
x + 11 divisible by 3 → x = 1, 4, 7
Answer: x = 1, 4, 7

vi. 42×8 divisible by 4?

Check last 2 digits: x8
Try values of x such that x8 divisible by 4:
08 ✔, 28 ✔, 48 ✔, 68 ✔, 88 ✔
x = 0, 2, 4, 6, 8
Answer: x = 0, 2, 4, 6, 8

vii. 9142x a multiple of 4?

Last two digits: 2x
Try values of x such that 2x divisible by 4:
20 ✔, 24 ✔, 28 ✔
x = 0, 4, 8
Answer: x = 0, 4, 8

viii. 7×34 divisible by 9?

Sum = 7 + x + 3 + 4 = x + 14
x + 14 divisible by 9 → x = 4
Answer: x = 4

ix. 5×555 a multiple of 9?

Sum = 5 + x + 5 + 5 + 5 = x + 20
x + 20 divisible by 9 → x = 7 (27 ✔)
Answer: x = 7

x. 3×2 divisible by 11?

Rule: Alternate sum = 3 − x + 2 = (5 − x)
(5 − x) divisible by 11 → x = -6 ❌ Not valid
Try x = 5 → 5 − 5 = 0 ✔
Answer: x = 5

xi. 5×2 a multiple of 11?

Alternate sum = 5 − x + 2 = (7 − x)
7 − x divisible by 11 → x = -4 ❌
Only valid value: x = 7 → 7 − 7 = 0 ✔
Answer: x = 7